The cruising speed of an airplane is 150 miles per hour...

The cruising speed of an airplane is 150 miles per hour (relative to the ground). You plan to hire the plane for a 3 -hour sightseeing trip. You instruct the pilot to fly north as far as she can and still return to the airport at the end of the allotted time. (A) How far north should the pilot fly if the wind is blowing from the north at 30 miles per hour? (B) How far north should the pilot fly if there is no wind?

The cruising speed of an airplane is 150 miles per hour (relative to the ground). You plan to hire the plane for a 3 -hour sightseeing trip. You instruct the pilot to fly north as far as she can and still return to the airport at the end of the allotted time.
(A) How far north should the pilot fly if the wind is blowing from the north at 30 miles per hour?
(B) How far north should the pilot fly if there is no wind?

Key Concepts

Piecewise Trip Analysis

Many travel problems require breaking a trip into separate segments with distinct conditions, such as flying outbound versus returning. Each segment may have a different effective speed due to factors like wind. By analyzing each leg independently and then summing their respective times, you can form an equation that captures the overall time constraint, which can then be solved for the desired variable such as distance.

Time-Distance Analysis

This concept involves the fundamental relationship between time, speed, and distance, summarized by the formula Distance = Speed × Time. In problems involving different speeds for different segments of a trip, the total time is the sum of the times for each segment. This analysis is essential to solve for unknown distances or times when specific speed conditions apply along various legs of a journey.

Relative Motion

This concept deals with understanding how the motion of an object is affected by the movement of the medium in which it is traveling. In the context of flight, the airplane’s effective speed over the ground (ground speed) is influenced by the wind’s velocity, such that the wind can either add to or subtract from the airplane's speed depending on its direction relative to the airplane’s course.

Ground Speed

Ground speed is the actual speed at which an aircraft moves relative to the ground. It is calculated by vectorially adding the airplane's own speed (its airspeed) to the wind speed. When flying into a headwind, the effective ground speed decreases, and when flying with a tailwind, it increases. This concept is crucial for determining travel times and distances in flight planning.

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